160 work book

Precalculus WorkbookDeveloped by: Jerry Morris, Sonoma State University

A Companion to Functions Modeling Change

Connally, Hughes-Hallett, Gleason, et. al.

c©Wiley, 2007

Note to Students (Please Read): This workbook contains examples and exercisesthat will be referred to regularly during class. Please purchase or print out the rest of the workbookbefore our next class and bring it to class with you every day.

1. To Print Out the Workbook. Go to the web address below

http://www.sonoma.edu/users/m/morrisj/m160/frame.html

and click on the link “Math 160 Workbook”, which will open the workbook as a .pdf file. BE FORE-WARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK,SO SOME PRINTERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK. Ifyou do choose to try to print it, please leave yourself enough time to purchase the workbook beforeour next class class in case your printing attempt is unsuccessful.

2. To Purchase the Workbook. Go to Digi-Type, the print shop at 1726 E. Cotati Avenue (across fromcampus, in the strip mall behind the Seven-Eleven). Ask for the workbook for Math 160 - Precalculus.The copying charge will probably be between $10.00 and $20.00. You can also visit the Digi-Type

webpage (http://www.digi-type.com/) to order your workbook ahead of time for pick-up.

2 Sonoma State University

Table of ContentsChapter 1 – Functions, Lines, and ChangeChapter 1 Tools – Points and Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Section 1.1 – Functions and Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Section 1.2 – Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Sections 1.3 & 1.4 – Linear Functions and Their Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Section 1.5 – Geometric Properties of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Chapter 2 – Functions, Quadratics, and ConcavitySection 2.1 – Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Section 2.2 – Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Section 2.4 – Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Section 2.5 – Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Section 2.6 – Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

Chapter 3 – Exponential Functions

Chapter 3 Tools – Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Sections 3.1-3.3 – Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Section 3.4 – Continuous Growth and the Number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 4 – Logarithmic Functions

Section 4.1 – Logarithms and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Section 4.2 – Logarithms and Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Section 4.3 – The Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Chapter 5 – Transformations of Functions and Their Graphs

Sections 5.1-5.3 – Function Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Section 5.5 – Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59

Chapter 6 – Trigonometric Functions

Section 6.1 – Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Section 6.2 – The Sine and Cosine Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Section 6.3 – Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Section 6.4 Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Sections 6.4 & 6.5 – Sinusoidal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71Section 6.6 – Other Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Section 6.7 – Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Chapter 7 – TrigonometrySection 7.1 – The Laws of Sines and Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Section 7.2 – Using Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86

Chapter 8 – Compositions, Inverses, and Combinations of Functions

Section 8.1 – Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Section 8.2 – Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

Chapter 9 – Polynomial and Rational Functions

Section 9.1 – Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99Sections 9.2 & 9.3 – Poynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Section 9.4 & 9.5 – Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Preliminary Review Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111

Precalculus Workbook, Wiley, 2007 3

Chapter 1 Tools – Points and Linear Equations

Example 1. Solve2

2− t=

3

2− 2tfor t.

Example 2. Solve 2 =A+Bt

A−Btfor t.


Example 3. WriteA

C +Bt+

B

D − Ctas a single fraction.

Example 4. Solve4x+ 3y = 112x− y

3 = 0for x and y.


Section 1.1 – Functions and Function Notation

Definition. A function is a rule that takes certain values as inputs and assigns to each input value exactlyone output value.

Example. Let y =2

1 + x.

x 0 1 2 3y

Example.

t = time (in years) after the year 2000

w = number of San Francisco ’49er victories

t 0 1 2 3 4 5 6 7 8 9w 6 12 10 7 2 4 7 5 7 8

Observations:Example. Which of the graphs below represent y as a function of x?

Graph 1y

x

Graph 2y

x

Graph 3y

x


Example. A woman drives from Aberdeen to Webster, South Dakota, going through Groton on the way,traveling at a constant speed for the whole trip. (See map below).

40 miles

Aberdeen Groton Webster

20 miles

a. Sketch a graph of the woman’s distance from Webster as a function of time.

b. Sketch a graph of the woman’s distance from Groton as a function of time.


Section 1.2 – Rates of Change

Preliminary Example. The table to the right showsthe temperature, T, in Tucson, Arizona t hours after mid-night.

Question. When does the temperature decrease thefastest: between midnight and 3 a.m. or between 3 a.m.and 4 a.m.?

t (hours after midnight) 0 3 4T (temp. in ◦F) 85 76 70

Graphical Interpretation of Rate of Change

Definition. The average rate of change, or just rate of change of Q with respect to t is given by

a

Q

t

f

bAlternate Formula for Rate of Change: The average rate of change of a function Q = f(t) on theinterval a ≤ t ≤ b is given by the following formula:


Examples and Exercises

1. Let f(x) = 4− x2. Find the average rate of change of f(x) on each of the following intervals.(a) 0 ≤ x ≤ 2 (b) 2 ≤ x ≤ 4 (c) b ≤ x ≤ 2b

2. To the right, you are given a graph of the amount, Q, ofa radioactive substance remaining after t years. Only thet-axis has been labeled. Use the graph to give a practical

interpretation of each of the three quantities that follow.A practical interpretation is an explanation of meaningusing everyday language.

2t (yrs)

Q (grams)

1 3

a. f(1)

b. f(3)

c.f(3)− f(1)

3− 1


3. Two cars travel for 5 hours along Inter-state 5. A South Dakotan in a 1983 ChevyCaprice travels 300 miles, always at aconstant speed. A Californian in a 2009Porsche travels 400 miles, but at varyingspeeds (see graph to the right).

t (hours)1 32 4 5

d (miles)400

300

200

100

(a) On the axes above, sketch a graph of the distance traveled by the South Dakotan as a function of time.

(b) Compute the average velocity of each car over the 5-hour trip.

(c) Does the Californian drive faster than the South Dakotan over the entire 5 hour interval? Justify youranswer!


Sections 1.3 & 1.4 – Linear Functions and Their Formulas

Preliminary Example. The cost, C, of your monthly phone bill consists of a $30 basic charge, plus $0.10 foreach minute of long distance calls.

(a) Complete the table below, and sketch a graph.

t 0 30 60 90 120 150C

(b) Compute the average rate of change of C over any time interval.

(c) Find a formula for C in terms of t.

(d) If your bill is $135, how long did you talk long distance?


Notes on Linear Functions:

1. If y = f(x) is a linear function, then y = mx+ b, where

2. If y = f(x) is linear, then input values produce output values.

Different forms for equations of lines:

Example 2. Find the slope and the y-intercept for each of the following linear functions.

(a) 3x+ 5y = 20

(b)x− y

5= 2



1. Let C = 20− 0.35t, where C is the cost of a case of apples (in dollars) t days after they were picked.

(a) Complete the table below:

t (days) 0 5 10 15

C (dollars)

(b) What was the initial cost of the case of apples?

(c) Find the average rate of change of C with respect to t. Explain in practical terms (i.e., in terms of costand apples) what this average rate of change means.

2. In parts (a) and (b) below, two different linear functions are described. Find a formula for each linearfunction, and write it in slope intercept form.

(a) The line passing through the points (1, 2) and(−1, 5).

(b)C 10 15 20 25F 50 59 68 77


3. According to one economic model, the demand for gasoline is a linear function of price. If the price ofgasoline is p = $3.10 per gallon, the quantity demanded in a fixed period of time is q = 65 gallons. If the priceis $3.50 per gallon, the quantity of gasoline demanded is 45 gallons for that period.

(a) Find a formula for q (demand) in terms of p (price).

(b) Explain the economic significance of the slope in the above formula. In other words, give a practicalinterpretation of the slope.

(c) According to this model, at what price is the gas so expensive that there is no demand?

(d) Explain the economic significance of the vertical intercept of your formula from part (a).

4. Look back at your answer to problem 2(b). You might recognize this answer as the formula for convertingCelsius temperatures to Fahrenheit temperatures. Use your formula to answer the following questions.

(a) Find C as a function of F.

(b) What Celsius temperature corresponds to 90◦F?

(c) Is there a number at which the two temperature scales agree?


Section 1.5 – Geometric Properties of Linear Functions

Example 1. You need to rent a car for one day and to compare the charges of 3 different companies. CompanyI charges $20 per day with an additional charge of $0.20 per mile. Company II charges $30 per day with anadditional charge of $0.10 per mile. Company III charges $70 per day with no additional mileage charge.

(a) For each company, find a formula for the cost, C, of driving a car m miles in one day. Then, graph the costfunctions for each company for 0 ≤ m ≤ 500. (Before you graph, try to choose a range of C values would beappropriate.)

(b) How many miles would you have to drive in order for Company II to be cheaper than Company I?


Example 2. Given below are the equations for five dif-ferent lines. Match each formula with its graph to theright.

• f(x) = 20 + 2x

• g(x) = 20 + 4x

• h(x) = 2x− 30

• u(x) = 60− x

• v(x) = 60− 2x

E

x

y A BC

D

Facts about the Line y = mx + b

1. The y-intercept, b (also called the vertical intercept), tells us where the line crosses the .

2. If m > 0, the line left to right. If m < 0, the line left to right.

3. The larger the value of |m| is, the the graph.

Parallel and Perpendicular Lines

Fact: Two lines (y = m1x+ b1 and y = m2x+ b2) are . . .

1. . . . parallel if

2. . . . perpendicular if


1. Consider the lines given in the figure to the right. Giventhat the slope of one of the lines is −2, find the exact

coordinates of the point of intersection of the two lines.(“Exact” means to leave your answers in fractional form.)

−2

x

y

2

3

2. Parts (a) and (b) below each describe a linear function. Find a formula for the linear function described ineach case.

(a) The line parallel to 2x−3y = 2 that goes throughthe point (1, 1).

(b) The line perpendicular to 2x− 3y = 2 that goesthrough the point (1, 1).


Section 2.1 – Input and Output

Preliminary Example. Complete each of the following.

x10 20

10

a b

f(x)

1. f(10) = .

2. If f(x) = 10, then x = .

3. f(a) = .

4. f(10)− f(6) = .



1. The following table shows the amount of garbage produced in the U.S. as reported by the EPA.

t (years: 1960 ≡ 60) 60 65 70 75 80 85 90G (millions of tons of garbage) 90 105 120 130 150 165 180

Consider the amount of garbage G as a function of time G = f(t). Include units with your answers.

(a) f(60) =

(b) f(75) =

(c) Solve f(t) = 165.

2. Given is the graph of the function v(t). It represents the velocity of a man riding his bike to the library andgoing back home after a little while. A negative velocity indicates that he is riding toward his house, awayfrom the library.

10

20

5 10 15 20 25 30 35 40 45 t (minutes)

v (mph)

15

5

-15

-10

-5

Evaluate and interpret:

(a) v(5) =

(b) v(40) =

(c) v(12)− v(7) =

Solve for t and interpret:

(d) v(t) = 5

(e) v(t) = −10

(f) v(t) = v(10)


3. Consider the functions given below.

(a) Let f(x) = x2 − 2x− 8.

i. Find f(0).

ii. Solve f(x) = 0.

(b) Let f(x) =1

x+ 2− 1

i. Find f(0).

ii. Solve f(x) = 0.

4. Let f(x) =x

x+ 1. Calculate and simplify f

(

1

t+ 1

)

, writing your final answer as a single fraction.


Section 2.2 – Domain and Range

Definition. If Q = f(t), then

1. The of f is the set of all input values, t, that yield a meaningful output value.

2. The of f is the corresponding set of all output values.

Example 1. Let A = f(r) be the area, in cm2, of a circle of radius r cm. Find the domain and the range of f.

Example 2. Find the domain and range of the function f(x) =√x+ 2.


1. For each of the following functions below, give the domain and the range.

f(x)

−4

−2

−4

2

4

2 4−2

g(x)

−4

−2

−4

2

4

2 4−2

2. Oakland Coliseum is capable of seating 63,026 fans. For each game, the amount of money that the Raider’sorganization makes is a function of the number of people, n, in attendance. If each ticket costs $30.00, findthe domain and range of this function. Sketch its graph.


3. Find the domain and range of each of the following functions.

(a) f(x) =√3x+ 7

(b) g(x) =1

(x − 1)2

(c) h(x) = x2 − x− 6

(d) k(x) =√x2 − x− 6


Section 2.4 – Inverse Functions

Preliminary Example. Recall the phone example from earlier, where a calling plan charged us a $30monthly service fee and then $0.10 per minute for long distance calls.

t 0 30 33 36 60C 30 33 33.30 33.60 36

For each of the following, fill in the blank and then give an interpretation of the entire statement.

(a) f(36) =

(b) f−1(36) =

(c) f−1( ) = 33



1. Use the two functions shown below to fill in the blanks to the right.f(x)

−4

−2

−4

2

4

2 4−2

x -6 -4 -2 0 2 4 6g(x) 2 0 3 7 6 1 5

(a) f(2) = (b) f−1(2) =

(c) g(0) = (d) g−1(0) =

(e) f(3) + 1 = (f) f−1(3) + 1 =

(g) f(3 + 1) = (h) f−1(3 + 1) =

(i) If g−1(x) = 0, then x = .

2. Let A = f(n) be the amount of periwinkle blue paint, in gallons, needed to paint n square feet of a house.Explain in practical terms what each of the following quantities represents. Use a complete sentence in eachcase.

(a) f(20)

(b) f−1(20)


3. If a cricket chirps R times per minute, then the outside temperature is given by T = f(R) = 14R + 40 degrees

Fahrenheit.

(a) Find a formula for the inverse function R = f−1(T ).

(b) Calculate and interpret f(50) and f−1(50).


Section 2.5 – Concavity

Definitions.

1. A function f(x) is called increasing if its graph from left to right. It is called decreasing if itsgraph from left to right.

2. A function f(x) is called concave up if its average rate of change increases from left to right.

3. A function f(x) is called concave down if its average rate of change decreases from left to right.

Describe the shape of the graph of a function f(x)that is concave up:

Describe the shape of the graph of a function f(x)that is concave down:

Example. Read the following description of a function. Then, decide whether the function is increasing ordecreasing. What does the scenario tell you about the concavity of the graph modeling it?

“When a new product is introduced, the number of people who usethe product increases slowly at first, and then the rate of increase isfaster (as more and more people learn about the product). Eventu-ally, the rate of increase slows down again (when most people whoare interested in the product are already using it).”


Example. Consider the following graphs of population, P, as a function of time, t.

(a)

t

P

(c)

t

P

(b)

t

P

(d)

t

P

Descriptions

(a) P is , and the rate of change of P is .

(b) P is , and the rate of change of P is .

(c) P is , and the rate of change of P is .

(d) P is , and the rate of change of P is .



1. Consider the functions shown below. Fill in the accompanying tables and then decide whether each functionis increasing or decreasing, and whether it is concave up or concave down.

(a) Description. This graph gives distance driven as a func-tion of time for a California driver.

t 0 2 3 5d∆d

∆t

t (hours)1 32 4 5

d (miles)400

300

200

100

(b) Description. This graph gives distance driven as a func-tion of time for a South Dakota driver.

t 0 2 3 5d∆d

∆t

t (hours)1 32 4 5

d (miles)400

300

200

100

(c) Description. This graph gives the amount of a decayingtwinkie as a function of time.

t 0 4 6 10A∆A

∆t

4

A (ounces)

t (years)2 4 6 8 10

1

2

3

(d) Description. This graph gives the amount of ice remain-ing in a melting ice cube as a function of time.

t 0 4 6 10A∆A

∆t

t (minutes)

A (ounces)

2 4 6 8 10

1

2

3

4


2. Decide whether each of the following functions are concave up, concave down, or neither.

x 0 1 2 3 4f(x) 1 3 6 10 20

x 0 1 2 3 4g(x) 10 9 7 4 0

p(x) = 3x+ 1

h(x)


Section 2.6 – Quadratic Functions

Definition. The general form of a quadratic function is given by

Some quadratic functions can also be written in the factored form

Note: The graph of a quadratic function is a .

Example. Find the zeros (if possible) of the following quadratic functions.

(a) f(x) = 2x2 + x− 3 (b) g(x) = x2 + 4x+ 2



1. Find (if possible), the zeros of the following quadratic functions.

(a) f(x) = x2 + 5x− 14(b) g(x) = x2 + 1

2. The height of a rock thrown into the air is given by h(t) = 40t− 16t2 feet, where t is measured in seconds.

(a) Calculate h(1) and give a practical interpretation of your answer.

(b) Calculate the zeros of h(t) and explain their meaning in the context of this problem.

(c) Solve the equation h(t) = 10 and explain the meaning of your solutions in the context of this problem.

(d) Use a graph of h(t) to estimate the maximum height reached by the stone. When, approximately, doesthe stone reach its maximum height? Is the function concave up or concave down?


Chapter 3 Tools – Exponents

Properties of Exponents

1. anam =

2.an

am=

3. (am)n =

4. (ab)n =

5.(a

b

)n

=

Caution!!

Some Definitions

(A) a0 = (B) a−n = (C) a1

n = (D) am

n =

Example 1. Without a calculator, simplify 9−1/2 +√0.01.

Example 2. Simplify√

xeye/2 + (xe)(xe)2.


Example 3. Simplify both of the following: (a)n−1a

a2(b)

n−1a+ 1

a2


Examples and ExercisesDirections. For problems 1-7, simplify. For problem 8, solve for x. You may need extra paper for your calculations.

1.(xy3)2

x0y5

2.(AB)4

A−1B−2

3.a3b−1

√a5/2

4. 2b−1(b2 + b)− 2

5.2M +M−1

1 + 2M−2

6. 3 3

√

t3 + 7(t9)1/3

7.2km3 + k2m

km−1

8. 81x = 3


Sections 3.1-3.3 – Exponential Functions

Example 1. The population of a rapidly-growing country starts at 5 million and increases by 10% each year.Complete the table below:

t (years) P, population ∆P, increase(in millions) in population (mil)

0

1

2

3

4

Definition. An exponential function Q = f(t) has the formula f(t) = abt, b > 0,

where

a =

b =

Note: b = 1 + r, where r is the decimal representation of the percent rate of change.

Example 2.

Description Growth Factor and Formula

The population, P, of ants in yourkitchen starts at 10 and increasesby 5% per day.The value, V, of a 1982 ChevyCaprice starts at $10000 and de-creases by 8% per year.

The air pressure, A, starts atmillibars at sea level (h =

0) and decreases by permile increase in elevation.

A = 960(0.8)h


Example 3. Analyze the functions f and g below. Which is linear? Which is exponential? Give a formula foreach function.

x 5 10 15 20 25f(x) 10 17 24 31 38g(x) 100 115 132.25 152.09 174.9


Comparison of Linear and Exponential Functions. If y = f(x) is given as a table of values,and if the x-values are equally spaced, then

1. f is linear if the of successive y-values is constant.

2. f is exponential if the of successive y-values is constant.

Example 4.

Below are the graphs of Q = 150(1.2)t, Q = 50(1.2)t,and Q = 100(1.2)t. Match each formula to the correctgraph.

Below are the graphs of Q = 50(1.2)t, Q = 50(0.6)t,Q = 50(0.8)t, and Q = 50(1.4)t. Match each formulato the correct graph.

Observations about the graph of Q = abt:



1. Suppose we start with 100 grams of a radioactive substance that decays by 20% per year. First, complete thetable below. Then, find a formula for the amount of the substance as a function of t and sketch a graph of thefunction.

t (years) 0 1 2 3 4Q (grams)

2. Suppose you invest $10000 in the year 2000 and that the investment earns 4.5% interest annually.

(a) Find a formula for the value of your investment, V, as a function of time.

(b) What will the investment be worth in 2010? in 2020? in 2030?


3. The populations of the planet Vulcan and the planet Romulus are recorded in 1980 and in 1990 according tothe table below. Also, assume that the population of Vulcan is growing exponentially and that the populationof Romulus is growing linearly.

Planet 1980 Population (billions) 1990 Population (billions)Vulcan 8 12Romulus 16 20

(a) Find two formulas; one for the population of Vulcan as a function of time and one for the population ofRomulus as a function of time. Let t = 0 denote the year 1980.

(b) Use your formulas to predict the population of both planets in the year 2000.

(c) According to your formula, in what year will the population of Vulcan reach 50 billion? Explain how yougot your answer.

(d) In what year does the population of Vulcan overtake the population of Romulus? Justify your answerwith an accurate graph and an explanation.


4. Find possible formulas for each of the two functions f and g described below.

x 0 2 4 6f(x) 2 2.5 3.125 3.90625

g(x)

x

y

H1,1��

3L

H-1, 2L

5. Consider the exponential graphs pictured below and the six constants a, b, c, d, p, and q.

(a) Which of these constants are definitely positive?

(b) Which of these constants are definitely between 0 and 1?

x

y

x

y=pq

y=cd

y=ab

x

x

(c) Which two of these constants are definitely equal?

(d) Which one of the following pairs of constants could be equal?

a and p b and d b and q d and q


Section 3.4 – Continuous Growth and the Number e

Preliminary Example. At the In-Your-Dreams Bank of America, all investments earn 100% interestannually. Suppose that you invest $1000 at a time that we will call month 0. Fill in the blanks below to comparewhat your investment will be worth 1 year later using various methods of interest compounding.

Month Compounded Compounded Compounded1 Time 2 Times 4 Times

0 $1000 $1000 $1000

12

34

567

89

1011

12

Alternative Formula for Exponential Functions. Given an exponential function Q = abt, it is possibleto rewrite Q as follows:

Q =

The constant k is then called the continuous growth rate of Q.

Notes:• If k > 0, then Q is increasing.

• If k < 0, then Q is decreasing.


Exercise Suppose that the population of a town starts at 5000 and grows at a continuous rate of 2% per year.

(a) Write a formula for the population of the town as a function of time, in years, after the starting point.

(b) What will the population of the town be after 10 years?

(c) By what percentage does the population of the town grow each year?


Section 4.1 – Logarithms and Their Properties

Definition. If x is a positive number, then

1. log x is

2. lnx is

In other words,

y = log x means that

y = lnx means that

Example 1. Calculate the following, exactly if possible.

(a) log 100

(b) log 0.1

(c) ln e

(d) ln(e2)

(e) log 5

Facts about Logarithms.

1. (a) log(10x) = ln(ex) =

(b) 10logx = eln x =

2. (a) log(ab) = ln(ab) =

(b) log(ab ) = ln(ab ) =

(c) log(bt) = ln(bt) =

3. (a) log 1 = ln 1 =

(b) log 10 = ln e =

Example 2. Solve each of the following equations for x.

(a) 5 · 4x = 25 (b) 5x4 = 25


Examples and Exercises1. Solve each of the following equations for x.

(a) 5 · 3x = 2 · 7x

(b) 10e4x+1 = 20

(c) a · bt = c · d2t

(d) 5x9 = 10

(e) e2x + e2x = 1

(f) ln(x+ 5) = 10


2. Simplify each of the following expressions.

(a) log(2A) + log(B)− log(AB) (b) ln(abt)− ln((ab)t)− ln a

3. Decide whether each of the following statements are true or false.

(a) ln(x + y) = lnx+ ln y

(b) ln(x + y) = (ln x)(ln y)

(c) ln(ab2) = ln a+ 2 ln b

(d) ln(abx) = ln a+ x ln b

(e) ln(1/a) = − lna


Section 4.2 – Logarithms and Exponential Models

Review: Two ways of writing exponential functions:

(1) Q = abt (2) Q = aekt

Example 1. Fill in the gaps in the chart below, assuming that t is measured in years:

Formula Growth or Decay Rate

Q = abt Q = aekt Per Year Continuous Per Year

Q = 6e−0.04t

Q = 5(1.2)t

Q = 10(0.91)t


Example 2. The population of a bacteria colony starts at 100 and grows by 30% per hour.

(a) Find a formula for the number of bacteria, P, after t hours.

(b) What is the doubling time for this population; that is, how long does it take the population to double in size?

(c) What is the continuous growth rate of the colony?


Definition. The half-life of a radioactive substance is the amount of time that it takes for

Example 3. The half-life of a Twinkie is 14 days.

(a) Find a formula for the amount of Twinkie left after t days.

(b) Find the daily decay rate of the Twinkie.



1. (Taken from Connally) Scientists observing owl and hawk populations collect the following data. Their initialcount for the owl population is 245 owls, and the population grows by 3% per year. They initially observe 63hawks, and this population doubles every 10 years.

(a) Find formulas for the size of the population of owls and hawks as functions of time.

(b) When will the populations be equal?


2. Find the half-lives of each of the following substances.

(a) Tritium, which decays at an annual rateof 5.471% per year.

(b) Vikinium, which decays at a con-tinuous rate of 10% per week.

3. If 17% of a radioactive substance decays in 5 hours, how long will it take until only 10% of a given sample ofthe substance remains?


Section 4.3 – The Logarithmic Function

1. Consider the functions f(x) = lnx and g(x) = log x.

(a) Complete the table below.

x 0.1 0.5 1 2 4 6 8 10ln xlog x

(b) Plug a few very small numbers x into lnx and log x (like 0.01, 0.001, etc.) What happens to the outputvalues of each function?

(c) If you plug in x = 0 or negative numbers for x, are lnx and log x defined? Explain.

(d) What is the domain of f(x) = lnx? What is the domain of g(x) = log x?

(e) Sketch a graph of f(x) = lnx below, choosing a reasonable scale on the x and y axes. Does f(x) haveany vertical asymptotes? Any horizontal asymptotes?

y

x


2. What is the domain of the following four functions?

(a) y = ln(x2)

(b) y = (lnx)2

(c) y = ln(lnx)

(d) y = ln(x− 3)

3. Consider the exponential functions f(x) = ex and g(x) = e−x. What are the domains of these two functions?Do they have any horizontal asymptotes? any vertical asymptotes?


Sections 5.1-5.3 – Function Transformations

Example 1. Consider the function f(x) = x2 − 4x+ 4.

Transformation Formula Graph Description

y = f(x) + 2

-4 -2 2 4

-4

-2

2

4

y = f(x)− 2

-4 -2 2 4

-4

-2

2

4

y = f(x+ 2)

-4 -2 2 4

-4

-2

2

4

y = f(x− 2)

-4 -2 2 4

-4

-2

2

4

y = f(−x)

-4 -2 2 4

-4

-2

2

4

y = −f(x)

-4 -2 2 4

-4

-2

2

4


Example 2. Let y = f(x) be the function whose graphis given to the right. Sketch the graphs of the transforma-tions y = f(x− 2), y = −2f(x), and y = f(−x). Then, fillin the entries in the table below.

x -4 -2 0 2 4 6f(x)

f(x− 2)−2f(x)f(−x)

4

2 4−6 −4 −2 6

−2

−4

2

Example 3. To the right, you are given the graph of a function f. Match eachgraph below to the appropriate transformation formula. Note that some transfor-mation formulas will not match any of the graphs.

(a) y = 2f(x)

(b) y = f(x) + 2

(c) y = 2− f(x)

(d) y = 2f(x) + 2

(e) y = f(x+ 2)

(f) y = f(−x)

(g) y = −f(x)

(h) y = 4f(x)

f



1. Write formulas for each of the following transformations of the function q(p) = p2 − p+ 1.

(a) q(p− 1) (b) q(p)− 1 (c) −2q(−p)

2. Let y = f(x) be the function whose graph is given below. Fill in the entries in the table below, and thensketch a graph of the transformations y = f(−x) and y = 1− 2f(x).

4

2 4−6 −4 −2 6

−2

−4

2

x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6f(x)f(−x)

1− 2f(x)


3. Given to the right is the graph of the function

y =

(

1

2

)x

. On the same set of axes, sketch the

graph of y =

(

1

2

)x−2

and y =

(

1

2

)x

− 2.

-4 -2 2 4

-2

2

4

4. Let H = f(t) be the temperature of a heated officebuilding t hours after midnight. (See diagram tothe right for a graph of f.) Write down a formulafor a new function that matches each story below.

(a) The manager decides that the temperatureshould be lowered by 5 degrees throughoutthe day.

t (hours)4 8 12 16 20 24

80

60

40

20

H (degrees F)

(b) The manager decides that employees should come to work 2 hours later and leave 2 hours later.


Definition Sketch

We say that a function is even if f(−x) = f(x)for all x in the domain of the function. In otherwords, an even function is symmetric about the

.

We say that a function is odd if f(−x) = −f(x)for all x in the domain of the function. In otherwords, an odd function is symmetric about the

.

5. Use algebra to show that f(x) = x4 − 2x2 + 1 is an even function and that g(x) = x3 − 5x is an odd function.


6. Given the graph of y = f(x) given to the right, sketch thegraph of the following related functions: f(x)

(a) y = −f(x+ 3) + 1f(x)

(b) y = 2− f(1− x)


Section 5.5 – The Family of Quadratic Functions

f(x) = x2

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

Information about Quadratic Functions

In general, a quadratic function f can be written in several different ways:

1. f(x) = ax2 + bx+ c (standard form, where a, b, and c are constant)

2. f(x) = a(x− r)(x − s) (factored form, where a, r, and s are constant)

3. f(x) = a(x− h)2 + k (vertex form, where a, h, and k are constant)

Notes.

• The graph of a quadratic function is called a .

• In factored form, the numbers r and s represent the of f.

• In vertex form, the point (h, k) is called the of the parabola. The axis of symmetry is the line

. The graph opens upward if and downward if .


Example 1. Find the vertex of y = x2 − 8x− 27 by completing the square.

Example 2. Find the vertex of y = 2x2 + 7x+ 3 by completing the square.


1. For each of the following, complete the square in order to find the vertex.

(a) y = x2 − 40x+ 1


(b) y = 2x2 + 12x+ 3

2. Find a formula for the quadratic function shown below. Also find the vertex of the function.

-1 2x

1

y


3. A parabola has its vertex at the point (2, 3) and goes through the point (6, 11). Find a formula for theparabola.

4. (Taken from Connally) A tomato is thrown vertically into the air at time t = 0. Its height, d(t) (in feet),above the ground at time t (in seconds) is given by d(t) = −16t2 + 48t.

(a) Find t when d(t) = 0. What is happening to the tomato the first time that d(t) = 0? The second time?

(b) When does the tomato reach its maximum height? How high is the tomato’s maximum height?


Section 6.1 – Periodic Functions

Preliminary Example. The Brown County FerrisWheel has diameter 30 meters and completes one full rev-olution every two minutes. When you are at the lowestpoint on the wheel, you are still 5 meters above the ground.Assuming you board the ride at t = 0 seconds, sketch agraph of your height, h = f(t), as a function of time.

60 120 180 240t

10

20

30

40h

Definition. A function f is called periodic if its output values repeat at regular intervals. Graphically,this means that if the graph of f is shifted horizontally by p units, the new graph is identical to the original.Given a periodic function f :

1. The period is the horizontal distance that it takes for the graph to complete one full cycle. That is, ifp is the period, then f(t+ p) = f(t).

2. The midline is the horizontal line midway between the function’s maximum and minimum outputvalues.

3. The amplitude is the vertical distance between the function’s maximum value and the midline.

Example 1. What are the amplitude, midline and period of the function h = f(t) from the preliminary example?



1. The function given below models the height, h, in feet, of the tide above (or below) mean sea level t hoursafter midnight.

(a) Is the tide rising or falling at 1:00 a.m.?

(b) When does low tide occur?

6 12 18 24t

-20

-10

10

20h

(c) What is the amplitude of the function? Give a practical interpretation of your answer.

(d) What is the midline of the function? Give a practical interpretation of your answer.

2. Which of the following functions are periodic? For those that are, what is the period?

1 2 3 4 5 6

-3

-2

-1

1

2

3

-4 -2 2 4

-4

-2

2

4

-6 -4 -2 2 4 6

0.5

1

1.5

4

4

−4

−4


Section 6.2 – The Sine and Cosine Functions

Angle Measurement in Circles

• Angles start from the positive x-axis.

• Counterclockwise defined to be positive.

x

y

Definition. The unit circle is the term used to describea circle that has its center at the origin and has radiusequal to 1. The cosine and sine functions are then definedas described below.

(−1,0) (1,0)

(0,1)

(0,−1)

Example 1. On the unit circle to the right, the angles10◦, 20◦, 30◦, etc., are indicated by black dots on thecircle. Use this diagram to estimate each of the following:

(a) cos(30◦) =

(b) sin(150◦) =

(c) cos(270◦) = H1,0L

H0,1L

Example 2.

(a) Find an angle θ between 0◦ and 360◦ that has the same sine as 40◦.

(b) Find an angle θ between 0◦ and 360◦ that has the same cosine as 40◦.


Theorem. Consider a circle of radius r centered at the origin. Thenthe x and y coordinates of a point on this circle are given by the followingformulas:

(−r, 0) (r, 0)

(0, −r)

(0, r)


1. Use the unit circle to the right to estimate each of thefollowing quantities to the nearest 0.05 of a unit.

(a) sin(90◦) = (b) cos(90◦) =

(c) sin(180◦) = (d) cos(180◦) =

(e) cos(45◦) = (f) sin(−90◦) =

(g) cos(70◦) = (h) sin(190◦) =

(i) sin(110◦) = (j) cos(110◦) =

H1,0L

H0,1L

2. For each of the following, fill in the blank with an angle between 0◦ and 360◦, different from the first one, thatmakes the statement true.

(a) sin(20◦) = sin( ) (b) sin(70◦) = sin( ) (c) sin(225◦) = sin( )

(d) cos(20◦) = cos( ) (e) cos(70◦) = cos( ) (f) cos(225◦) = cos( )

3. Given to the right is a unit circle. Fill in the blanks withthe correct answer in terms of a or b.

(a) sin(θ + 360◦) =

(b) sin(θ + 180◦) =

(c) cos(180◦ − θ) =

(d) sin(180◦ − θ) =

(e) cos(360◦ − θ) =

(f) sin(360◦ − θ) =

(g) sin(90◦ − θ) =

(a,b)

x

y

θ1


4. Use your calculator to find the coordinates of the point P at the given angle on a circle of radius 4 centeredat the origin.

(a) 70◦ (b) 255◦

Section 6.3 – Radian Measure

Definition. An angle of 1 radian is defined to bethe angle, in the counterclockwise direction, at thecenter of a unit circle which spans an arc of length 1.

Conversion Factors:

Degrees × −→ Radians

Radians × −→ Degrees

1

Example 2. Convert each of the following angles from radians to degrees or from degrees to radians. An anglemeasure is assumed to be in radians if the degree symbol is not indicated after it.

(a) 30◦ (b) 3π2 (c) 1.4

Example 3. On the unit circle to the right, label the indicated “com-mon” angles with their degree and radian measures.


Theorem. The arc length, s, spanned in a circle of radius r by anangle of θ radians, 0 ≤ θ ≤ 2π, is given by


1. In the pictures below, you are given the radius of a circle and the length of a circular arc cut off by an angleθ. Find the degree and radian measure of θ.

2θ4

4θ8

2. In the pictures below, find the length of the arc cut off by each angle.

2

ο803

2π/3

3. A satellite orbiting the earth in a circular path stays at a constant altitude of 100 kilometers throughout itsorbit. Given that the radius of the earth is 6370 kilometers, find the distance that the satellite travels incompleting 70% of one complete orbit.


4. An ant starts at the point (0,3) on a circle of radius 3 (centered at the origin) and walks 2 unitscounterclockwise along the arc of the circle. Find the x and the y coordinates of where the ant ends up.

Section 6.4 Supplement

Preliminary Example. Use the unit circles and corresponding triangles below to find the exact value of thesine and cosine of the special angles 30◦, 45◦, and 60◦.

1

45o

Figure 1.

o

o301

30

Figure 2.

o

1

60

Figure 3.


The Unit Circle

θ 0 π6

π4

π3

π2

2π3

3π4

5π6 π

θ 0◦ 30◦ 45◦ 60◦ 90◦ 120◦ 135◦ 150◦ 180◦

cos θ

sin θ

θ 7π6

5π4

4π3

3π2

5π3

7π4

11π6 2π

θ 210◦ 225◦ 240◦ 270◦ 300◦ 315◦ 330◦ 360◦

cos θ

sin θ

-Π��2

Π��2

Π 3 Π��2

2 ΠΘ

-1

-0.5

0.5

1


Sections 6.4 and 6.5 – Sinusoidal Functions

Directions. Make sure that your graphing calculator is set in radian mode.

Function Effect on y = sinxy = 2 sinx

y = sinx+ 2

y = sin(x+ 2)

y = sin(2x)

B y = sin(Bx) Period1 y = sinx2 y = sin(2x)4 y = sin(4x)1/2 y = sin(x/2)B y = sin(Bx)

SummaryFor the sinusoidal functions y = A sin(B(x− h)) + k and y = A cos(B(x− h)) + k:

1. Amplitude =

2. Period =

3. Horizontal Shift =

4. Midline:

Definition. A function is called sinusoidal if it is a transformation of a sine or a cosine function.

Primary Goal in Section 6.5. Find formulas for sinusoidal functions given graphs, tables, or verbaldescriptions of the functions.

Helpful Hints in Finding Formulas for Sinusoidal Functions

1. If selected starting point occurs at the midline of the graph, use the sine function.

2. If selected starting point occurs at the maximum or minimum value of the graph, use the cosinefunction.

3. Changing the sign of the constant “A” reflects the graph of a sinusoidal function about its midline.

Example 1. Let y = 2 sin(2x− π) + 2. Find the amplitude, period, midline, and horizontal shift of this function.



1. Find a possible formula for each of the following sinusoidal functions.

-Π Π 2 Π 3 Π 4 Π

3

6

-4 -2 2 4 6 8

-3

-2

-1

1

2 4

-3

3

Π��7

-0.8

0.8

Π

-3

3

2 11

2

10


2. For each of the following, find the amplitude, the period, the horizontal shift, and the midline.

(a) y = 2 cos(πx + 2π3 )− 1

(b) y = 3− sin(2x− 7π)

3. A population of animals oscillates annually from a low of 1300 on January 1st to a high of 2200 on July 1st,and back to a low of 1300 on the following January. Assume that the population is well-approximated by asine or a cosine function.

(a) Find a formula for the population, P, as a function of time, t. Let t represent the number of months afterJanuary 1st. (Hint. First, make a rough sketch of the population, and use the sketch to find theamplitude, period, and midline.)

(b) Estimate the animal population on May 15th.

(c) On what dates will the animal population be halfway between the maximum and the minimumpopulations?


Section 6.6 – Other Trigonometric Functions

cos θ =

sin θ =

tan θ = cot θ =

sec θ = csc θ =

θ

1

(x, y)

Example 1. Suppose that cos θ = 25 and that θ is in the 4th quadrant. Find sin θ and tan θ exactly.

Example 2. Find exact values for each of the following:

(a) tan(

π6

)

(b) tan(

π4

)

(c) tan(

π2

)


Reference Angles

Definition. The reference angle associated with an angle θ is the acute angle (having positive measure)formed by the x-axis and the terminal side of the angle θ.

Example. For each of the following angles, sketch the angle and find the reference angle.

(1) θ = 300◦ (2) θ = 4π3

(3) θ = 135◦ (4) θ = 7π6

Key Fact. If θ is any angle and θ′ is the reference angle, then

sin θ′ = ± sin θ

cos θ′ = ± cos θ

tan θ′ = ± tan θ

csc θ′ = ± csc θ

sec θ′ = ± sec θ

cot θ′ = ± cot θ,

where the correct sign must be chosen based on the quadrant of the angle θ.

Exercise. Return to the previous example and find the exact value of the sine and the cosine of each angle.



1. Suppose that sin θ = − 34 and that 3π

2 ≤ θ ≤ 2π. Find the exact values of cos θ and sec θ.

2. Suppose that csc θ = x2 and that θ lies in the 2nd quadrant. Find expressions for cos θ and tan θ in terms of x.


3. Given to the right is a circle of radius 2 feet (notdrawn to scale). The length of the circular arcs is 2.6 feet. Find the lengths of the segmentslabeled u, v, and w. Give all answers rounded tothe nearest 0.001.

θu

sw

v2


Section 6.7 – Inverse Trigonometric Functions

Preliminary Idea.

sin(π/6) = 1/2 means the same thing as .

Definition.

1. sin−1 x is the angle between −π2 and π

2 whose sine is x.

2. tan−1 x is the angle between −π2 and π

2 whose tangent is x.

3. cos−1 x is the angle between 0 and π whose cosine is x.

Note. “sin−1 x”, “cos−1 x”, and “tan−1 x” can also be written as “arcsinx”, “arccosx”, and “arctanx”,respectively.

Example 1. Calculate each of the following exactly.

1. cos−1

(√3

2

)

=

2. sin−1

(√2

2

)

=

3. tan−1(√3) =

4. sin−1(−1) =

Example 2. Use the graph to estimate, to the nearest 0.1,all solutions to the equation sinx = − 1

2 that lie between 0 and2π. Then, find the solutions exactly.

-1 1 2 3 4 5 6x

-1

-0.5

0.5

1y = sin x



1. Solve each of the following trigonometric equations, giving all solutions between 0 and 2π. Give exact

answers whenever possible.

(a) sin θ =√32

(b) tan θ = −0.3

(c) cos θ = − 12

(d) sin θ = 0.7


2. Find all solutions to 2 sinx cos x+ cosx = 0 that lie between 0 and 2π. Give your answers exactly.

3. Use the graph to the right to estimate the solu-tions to the equation cosx = 0.8 that lie between0 and 2π. Then, use reference angles to find moreaccurate estimates of your solutions.

y = cosx

-1 1 2 3 4 5 6x

-1

-0.5

0.5

1y


x sin x

0 0π/6 1/2

π/4√2/2

π/3√3/2

π/2 1

x sin−1 x

-1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

O

PQ

R S

x tan x

0 0

π/6√3/3

π/4 1

π/3√3

π/2 undefined

x tan−1 x

-2 -1 1 2

-2

-1

1

2

O

P

Q

R


Section 7.1 – The Laws of Sines and Cosines

Right Triangles

sin θ =

cos θ =

tan θ =θ

(r,0)

(0,r)

Warmup example. A kite flyer wondered how high her kite was flying. She used a protractor to measure anangle of 40◦ from level ground to the kite string. If she used a full 100-yard spool of string, how high is the kite?

General Triangles: The following formulas hold for any triangle, labeled as shown below.

Law of Sines:

Law of Cosines:

A

BC

a

cb

General Rule. The Law of Cosines can be used when 2 sides of a triangle and the angle in between the sides areknown.


Example. Find all possible triangles with a = 3, b = 4, and A = 35◦.



1. Two fire stations are located 25 miles apart, at points A and B. There is a forest fire at point C. If∠CAB = 54◦ and ∠CBA = 58◦, which fire station is closer? How much closer? (Taken from Connally, et. al.)

2. A triangular park is bordered on the south by a 1.7-mile stretch of highway and on the northwest by a 4-milestretch of railroad track, where 33◦ is the measure of the acute angle between the highway and the railroadtracks. As a part of a community improvement project, the city wants to fence the third side of the park andseed the park with grass.

(a) How much fence is needed for the third side of the park?

(b) What is the degree measure of the angle on the southeast side of the park?

(c) For how much total area will they need grass seed?


3. To measure the height of the Eiffel Tower in Paris, a per-son stands away from the base and measures the angle ofelevation to the top of the tower to be 60◦. Moving 210feet closer, the angle of elevation to the top of the tower is70◦. How tall is the Eiffel Tower? (Taken from Connally,

et. al.)

4. Two points P and T are on opposite sides of a river (seesketch to the right). From P to another point R on thesame side is 300 feet. Angles PRT and RPT are found tobe 20◦ and 120◦, respectively. (Taken from Cohen.)

(a) Compute the distance from P to T.

PR

T

(b) Assuming that the river is reasonably straight, calculate the shortest distance across the river.


Section 7.2 – Using Trigonometric Identities

Preliminary Example. Discuss the difference between the equation (a) sin θ = cos θ and the equation (b)sin(2θ) = 2 sin θ cos θ

Example 1. Are any of the following identities? If so, prove them algebraically.

(a) cos(

1x

)

= 1cosx (b) 2 tanx cos2 x = sin(2x)

Some Trigonometric Identities

sin(2θ) = 2 sin θ cos θ

cos(2θ) = 1− 2 sin2 θ

cos(2θ) = 2 cos2 θ − 1

cos(2θ) = cos2 θ − sin2 θ

sin2 θ + cos2 θ = 1

cos2 θ = 1− sin2 θ

sin2 θ = 1− cos2 θ

sin(−θ) = − sin θ

cos(−θ) = cos θ

tan(−θ) = − tan θ


Example 2. Rewrite the expression sec θ − cos θ so that your final answer is a product of two trig functions.

Example 3. Solve the equation 2 sin2 θ = 3 cos θ + 3 for 0 ≤ θ ≤ 2π.



1. Rewrite each of the following as indicated by the instructions.

Part Starting Expression Rewriting Instructions(a) sinA(cscA− sinA) Simplify so that your final answer is a single trig function raised to

a power, with no fractions.

(b)1− cos2 θ

cos θSimplify so that your final answer has no fractions and is a productof two trig functions, with no fractions.

(c)cos(2t)

cos t+ sin tSimplify so that your final answer is a difference of two trig func-tions with no fractions.(Hint: Use the identity cos(2t) = cos2 t− sin2 t and then factor.)


2. Simplifysin θ

1 + cos θ+

1 + cos θ

sin θso that your final answer is a constant multiple of just one trig function, with

no fractions.

3. By starting with one side and showing that it is equal to the other side, prove the following trigonometricidentity:

sin t

1− cos t=

1 + cos t

sin t


4. Given the right triangle to the right, write each of the fol-lowing quantities in terms of h.

Note. Asking you to “write something in terms of h” isNOT asking you to solve for h. It is asking you to rewritethe quantity that you are given so that h is the only un-known in your answer.

h1

θ(a) sin θ (b) cos θ

(c) cos(π2 − θ) (d) sin(2θ)

(e) cos(sin−1 h)


Section 8.1 – Function Composition

The function h(t) = f(g(t)) is called the composition of f with g. The function h is defined by using theoutput of the function g as the input of f.

Example 1. Complete the table below.

t 0 1 2 3f(t) 2 3 1 1g(t) 3 1 2 0

f(g(t)) 3 2g(f(t)) 0 1

Example 2. Let f(x) = x2 − 1, g(x) =2x2

x− 1, and p(x) =

√x. Find and simplify each of the following.

(a) g(f(x)) (b) p(g(x2)) (c)f(x+ h)− f(x)

h


Example 3. For the function f(x) = (x3 + 1)2, find functions u(x) and v(x) such that f(x) = u(v(x)).


1. Given to the right are the graphs of two functions, f and g. Use the graphsto estimate each of the following.

(a) g(f(0)) = (b) f(g(0)) =

(c) f(g(3)) = (d) g(g(4)) =

(e) f(f(1)) =

g(x)−4 4

−4

4

f(x)

2. For each of the following functions f(x), find functions u(x) and v(x) such that f(x) = u(v(x)).

(a)√1 + x (b) sin(x3 + 1) cos(x3 + 1)


(c) 32x+1 (d)1

1 + 2x

3. Let f(x) =1

1 + 2x.

(a) Solve f(x+ 1) = 4 for x.

(b) Solve f(x) + 1 = 4 for x.

(c) Calculate f(f(x)) and simplify your answer.


4. For each of the following functions, calculate

f(x+ h)− f(x)

h

and simplify your answers.

(a) f(x) = x2 + 2x+ 1 (b) f(x) = 1x (c) f(x) = 3x+ 1


Section 8.2 – Inverse Functions

Definition. Suppose Q = f(t) is a function with the property that each value of Q determines exactly onevalue of t. Then f has an inverse function, f−1, and

f−1(Q) = t if and only if Q = f(t).

If a function has an inverse, it is said to be invertible.

Example 1. Given below are values for a function Q = f(t). Fill in the corresponding table for t = f−1(Q).

t 0 1 2 3 4

f(t) 2 5 7 8 11

Q 2 5 7 8 11

f−1(Q)

Question. Does the function f(x) = x2 have an inversefunction?

f(x) = x2

-2 -1 1 2

1

2

3

4

Horizontal Line Test. A function f has an inverse function if and only if the graph of f intersectsany horizontal line at most once. In other words, if any horizontal line touches the graph of f in more thanone place, then f is not invertible.


Example 2. Suppose B = f(t) = 5(1.04)t, where B is the balance in a bank account, in thousands of dollars,after t years.

(a) Find a formula for the inverse function of f.

(b) Compute each of the following and interpret them practically: (i) f(20) (ii) f−1(20)



1. Find a formula for the inverse function of each of the following functions.

(a) f(x) =x− 1

x+ 1

(b) g(x) = ln(3− x)

2. Given to the right is the graph of the functions f(x) and g(x). Usethe function to estimate each of the following.

(a) f(2) = (b) f−1(2) =

(c) f−1(g(−1)) = (d) g−1(f(3)) =

g(x)

4

4

−4

−4

f(x)

(e) Rank the following quantities in order from smallest to largest: f(1), f(−2), f−1(1), f−1(−2), 0


3. Let f(x) = 10e(x−1)/2 and g(x) = 2 lnx− 2 ln 10 + 1. Show that g(x) is the inverse function of f(x).

4. Let f(t) represent the amount of a radioactive substance, in grams, that remains after t hours have passed.Explain the difference between the quantities f(8) and f−1(8) in the context of this problem.


Section 9.1 – Power Functions

Definition. A power function is a function of the form f(x) = kxp, where k and p are constants.

I. Positive Integer Powers. Match the following functions to the appropriate graphs below:y = x2, y = x3, y = x4, y = x5

-2 -1 1 2 -1 1

II. Negative Integer Powers. Match the following functions to the appropriate graphs below:y = x−2, y = x−3, y = x−4, y = x−5

-1 1 -1 1

III. Positive Fractional Powers. Match the following functions to the appropriate graphs below:y = x1/2, y = x1/3

1


Example 1. Find a formula for the power function that goes through the points (4, 32) and (14 ,12 ).

Definition.

1. A quantity y is called directly proportional to a power of x if , where k and n areconstants.

2. A quantity y is called inversely proportional to a power of x if , where k and n areconstants.

Example 2. Write formulas that represent the following statements.

(a) The pressure, P, of a gas is inversely proportional to its volume, V.

(b) The work done, W, in stretching a spring is directly proportional to the square of the distance, d, that it isstretched.

(c) The distance, d, of an object away from a planet is inversely proportional to the square root of thegravitational force, F, that the planet exerts on the object.


Example 3. (Adapted from Connally) The blood circulation time (t) of a mammal is directlyproportional to the 4th root of its mass (m). If a hippo having mass 2520 kilograms takes 123 seconds for its bloodto circulate, how long will it take for the blood of a lion with body mass 180 kg to circulate?


1. Find a formula for the power function g(x) described bythe table of values below. Be as accurate as you can withyour rounding.

x 2 3 4 5g(x) 4.5948 7.4744 10.5561 13.7973


2. The pressure, P, exerted by a sample of hydrogen gas is inversely proportional to the volume, V, in thesample. A sample of hydrogen gas in a 2-liter container exerts a pressure of 1.5 atmospheres. How muchpressure does the sample of gas exert if the size of the container is cut in half?

Section 9.2/9.3 – Polynomials

Definition. A polynomial is a function of the form

y = p(x) = anxn + an−1x

n−1 + · · ·+ a1x+ a0,

where n is a positive integer and a0, a1, . . . , an are all constants. The integer n is called the degree of thepolynomial.

Fact 1. A polynomial of degree n can have at most n− 1 “turnaround” points.

Fact 2. As x → ∞ and x → −∞, the highest power of x “takes over.” (Note. The symbol “→” means“approaches.”)


Fact 3. When a polynomial touches but does not cross the x axis at x = a, the factored form of the polynomialwill have an even number of (x− a) factors.

Example. Consider the polynomial

p(x) = (x+ 3)(x+ 2)2(x+ 1)(x− 1)(x− 2)2(x − 3)2,

whose graph is shown to the right.

-3 -2 -1 1 2 3 4

-400

-200

200

400

600

Definition. Let p(x) = anxn + an−1x

n−1 + · · · + a1x + a0 be a polynomial such that an 6= 0. Then thenumber an is called the leading coefficient of p, and the number a0 is called the constant coefficient of p.

Example. Find the leading coefficient and the constant coefficient of each of the following.

1. p(x) = 3x4 − 5x2 + 6x− 1

2. q(x) = x2(x− 3)

3. r(x) = (2x− 3)2(x+ 4)


Examples and ExercisesEach of the following gives the graph of a polynomial. Find a possible formula for each polynomial. In some cases,more than one answer is possible.

1.

-1 3x

-3

y 2.

-3-2 2x

-24

y

3.

-6-4 4x

y 4.

-3-2 2x

y

5.

-4 1 3x

y

H-2,-4L

6.

-1 1 3x

yH2, 1L


For problems 7 through 10, answer each of the following questions. Use a graphing calculator where appropriate.

a. How many roots (zeros) does the polynomial have?b. How many turning points does the polynomial have?

7. y = 2x+ 3

8. y = x2 − x− 2

9. y = x3 − 2x2 − x+ 2

10. y = 5x2 + 4

For problems 11 through 15, answer the following questions about the given polynomial:

a. What is its degree?b. What is its leading coefficient?c. What is its constant coefficient?d. What are the roots of the polynomial? First, give your answer(s) in exact form; then, give decimal

approximations if appropriate.

11. p(x) = x2 − 3x− 28

12. p(x) = 8− 7x

13. p(x) = x(2 + 4x− x2)


14. p(x) = 2x2 + 4

15. p(x) = (x− 3)(x+ 5)(x− 37)(2x+ 4)x2

For problems 16 through 18, answer the following questions about the given polynomial:

a. What happens to the output values for extremely positive values of x?b. What happens to the output values for extremely negative values of x?

16. p(x) = −2x3 + 6x− 2

17. p(x) = 2x− x2

18. p(x) = −x6 − x− 2

19. For each of the following, give a formula for a polynomial with the indicated properties.

a. A sixth degree polynomial with 6 roots.

b. A sixth degree polynomial with no roots.


Sections 9.4 and 9.5 – Rational Functions

Definition. A rational function is a function r(x) of the form r(x) =p(x)

q(x), where p(x) and q(x) are

polynomials. In other words, a rational function is a polynomial divided by a polynomial.

Example. Let f(x) =3x2 + 2x− 1

2x2 + 1. First, fill in the

table to the right for the function f(x). Then, sketch agraph of f(x) in the space below.

x 1 10 100 1000 10000f(x)

Example. Algebraically check each of the following for horizontal asymptotes.

(a) f(x) =3x2 + 2x− 1

2x2 + 1(b) g(x) =

2x+ 4

2x2 + 1(c) h(x) =

x6 + 5x3 − 2x2 + 1

x4 + 2


Finding horizontal asymptotes. Let f(x) = p(x)q(x) be a rational function.

1. If the degree of p(x) equals the degree of q(x), then f(x) has a horizontal asymptote at

y =leading coefficient of p(x)

leading coefficient of q(x).

2. If the degree of p(x) is less than the degree of q(x), then y = 0 is a horizontal asymptote.

3. If the degree of p(x) is greater than the degree of q(x), then f(x) has no horizontal asymptote.

Example. Let f(x) =x− 1

x+ 2. What happens when x = 1? What happens when x = −2? Graph this function for

−5 ≤ x ≤ 5 and −5 ≤ y ≤ 5.

x -1.9 -1.99 -1.999 -1.9999f(x)

x -2.1 -2.01 -2.001 -2.0001f(x)

Finding vertical asymptotes. Let f(x) = p(x)q(x) be a rational function. To find vertical asymptotes,

look at places where the denominator q(x) = 0.


Examples and ExercisesFor each of the following rational functions, find all horizontal and vertical asymptotes (if there are any), allx-intercepts (if there are any), and the y-intercept. Find exact and approximate values when possible.

1. f(x) =3x− 4

7x+ 1

2. f(x) =x2 + 10x+ 24

x2 − 2x+ 1

3. f(x) =2x3 + 1

x2 + x

4. f(x) =(x2 − 4)(x2 + 1)

x6

5. f(x) =2x+ 1

6x2 + 31x− 11


6. f(x) =2x2 − c

(x− c)(3x+ d), where c and d are constants, and c 6= 0.

7. f(x) =1

x− 3+

1

x− 5Hint: First, find a common denominator.

8. f(x) =x5 − 2x4 − 9x+ 18

8x3 + 2x2 − 3x

Hint: x5 − 2x4 − 9x+ 18 = x4(x − 2)− 9(x− 2)

9. f(x) =1

x− 1+

2

x+ 2+ 3

Hint: First, find a common denominator.


Math 107 Name:Algebra Review QuizFall 2010 Show your work for credit!

1. Solve each of the following equations for x, and circle your final answer.

(a) 2(x− 3) = 5x+ 2

(b) −2

3x = 2

(

4

3− x

6

)

2. Find the equation of the line that goes through the points (−1, 4) and (3, 3). Write your final answer in slopeintercept form, and circle it.

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